Mathematics: Aparabola has a focus of f(2, -0.5) and a directrix of y=-1.5 p(x,y) represents any point on the parabola, while d(x, -1.5) represents any point on the directrix. what are the steps required to find the equation of the parabola?

Aparabola has a focus of f(2, -0.5) and a directrix

Ответ:

negativechill

11.01.2022

Mathematics

The sketch of the parabola is attached below

We have the focus (a,b) = (2, -0.5)

The point

The directrix, c at y=-1.5

The steps to find the equation of the parabola are as follows

Step 1
Find the distance between the focus and the point P using Pythagoras. We have two coordinates; (2, -0.5)

and

.
We need the vertical and horizontal distances to find the hypotenuse (the diagram is shown in the second diagram).
The distance between the focus and point P is given by
$\sqrt{ (x-a)^{2}+ (y-b)^{2} }$

Step 2
Find the distance between the point P to the directrix

. It is a vertical distance between y and c, expressed as y-c

Step 3
The equation of parabola is then given as
$\sqrt{ (x-a)^{2}+ (y-b)^{2} }$ = y-c

$(x-a)^{2}+ (y-b)^{2}= (y-c)^{2}$ ⇒ substituting a, b and c
$(x-2)^{2}+ (y--0.5)^{2} = (y--1.5)^{2}$
$(x-2)^{2}+ (y+0.5)^{2}= (y+1.5)^{2}$ ⇒Rearranging and making